Revista Brasileira de Ensino de Fısica, v. 36, n. 2, 2309 (2014)www.sbfisica.org.br

A downward buoyant force experiment(Um experimento em que a forca de empuxo aponta para baixo)

F.M.S. Lima1, G.M. Venceslau1, G.T. Brasil2

1Instituto de Fısica, Universidade de Brasılia, Brasılia, DF, Brasil2Faculdade de Ciencia Exatas e Teconologia, Centro Universitario de Brasilia, Brasılia, DF, Brasil

Recebido em 17/9/2013; Aceito em 16/1/2014; Publicado em 11/5/2014

In hydrostatics, the Archimedes principle predicts an upward force whenever a body is submerged in a liquid.In contrast to common sense, this physical law is not free of exceptions, as for example when the body touchesthe container. This is more evident when a rectangular block less dense than the liquid rests on the bottom,with no liquid underneath it, a case in which a downward force is expected, according to a recent work by thefirst author. In the present work, we describe a simple, low-cost experiment which allows the detection of suchforce. This counterintuitive result shows the inadequacy of Archimedes’ principle for treating “contact” cases.Keywords: hydrostatics, Archimedes’ principle, Buoyant force.

Em hidrostatica, o princıpio de Arquimedes preve uma forca vertical para cima sempre que um corpo encontra-se submerso em um lıquido. Contrariamente ao senso comum, esta lei fısica nao esta livre de excecoes, como, porexemplo, quando o corpo esta em contato com o recipiente. Isto fica mais evidente quando um paralelepıpedomenos denso do que o lıquido encontra-se em repouso no fundo do recipiente, sem nenhum lıquido abaixo dele,um caso em que uma forca para baixo e esperada, de acordo com um trabalho recente do primeiro autor. Nestetrabalho nos apresentamos um experimento simples e de baixo custo que permite a deteccao de tal forca. Esteresultado contra-intuitivo mostra que o princıpio de Arquimedes e inadequado para casos em que ha contato docorpo com o recipiente.Palavras-chave: hidrostatica, princıpio de Arquimedes, forca de empuxo.

1. Introduction

In everyday life, it is easy to observe the buoyancy phe-nomenon for a solid body lighter than a liquid. Thecommon sense says that when a ‘light’ body is immersedin a liquid it will be pushed upwards until floating. Atschool, this kind of knowledge gains the status of a sci-entific law when we learn the celebrated Archimedes’Principle (AP) [1, 2]:

When a body is fully or partially submergedin a fluid, a buoyant force from the sur-rounding fluid acts on the body. This forceis directed upward and has a magnitudeequal to the weight of the fluid displacedby the body.

Our confidence in this law is indeed reinforced in in-troductory physics courses [1,2] or courses of fluid me-

chanics for engineers [3], in which AP is presented as alaw free of exceptions. However, when a body touchesany of the container’s walls the net hydrostatic forceis not necessarily vertical, as indicated for the smallerblock at the left of Fig. 1. This is indeed more evident inthe so-called ‘bottom case’, also indicated in Fig. 1, inwhich the net hydrostatic force points downward. Thiscounterintuitive situation was firstly reported in an ex-perimental work by Jones and Gordon, in which a pieceof cork glued to a thin aluminum disc was pressed on alarge aluminum block, both immersed in water [4].2 Onnoting that the metallic surfaces stuck together even af-ter the water was removed (i.e., in air), Ray and John-son soon complemented this experiment by placing theobjects into an evacuated cell and verifying that theywere effortless separated, showing that the force hold-ing the surfaces together is not adhesive (i.e., inter-molecular attraction between the surfaces), but due toatmospheric pressure [5].

1E-mail: fabio@fis.unb.br.

2In the words of those authors, “this experiment generates a great deal of interest, particularly if the students are drawn into adiscussion to predict the result”.

Copyright by the Sociedade Brasileira de Fısica. Printed in Brazil.

2309-2 Lima et al.

g

H

Fb

Fb

z

h

Figure 1 - The hydrostatic forces acting on rectangular blocks incontact to the walls of a container. The larger block, with heighth , represents the ‘bottom’ case. The arrows indicate the pressureforces exerted by the liquid on the surface of each block. The netforce exerted by the liquid in each block — i.e., the ‘buoyant’force, as defined in the text —, is represented by Fb. As usual,the z-axis points upwards.

In a more recent work, Valiyov and Yegorenkov dis-cussed the existence of a BF in the ‘bottom case’ andsuggest an experiment in which a table-tennis ball isattached to a thin glass plate which is placed on a largeglass block [6]. When the set is fully submerged in wa-ter the ball would remain below the waterline, whichled those authors to conclude (correctly, in our opin-ion) that AP as stated above is deficient. Their articleis followed by a comment by Kibble, who argues thatthese experiments are out of the scope of AP becausethe word ‘immersed’ would mean completely surroundedby the liquid [7]. However, his opinion is deficient be-cause it excludes the case of a solid body floating in adenser liquid with its emerged portion exposed to air(or vacuum), a case in which AP of course works! Kib-ble indeed points out that, due to an adhesive force,the thin glass plate and the large glass block could betreated as a single object for which the weight is greaterthan the BF, which is incorrect since adhesive forceswere ruled out by Ray and Johnson’s experiment [5].

In an experimental work by Bierman and Kincanon(2003), the validity of AP as stated in both the origi-nal and modern texts is reconsidered [8]. By cutting ahole in the bottom of a rubber-lined bucket and puttingan aluminum block over the hole, they overcame theproblem of water seepage. They then measured thedownward force exerted by the liquid on the block fordifferent depths using a PASCO sensor, showing thatit increases with depth, instead of being a constant (aspredicted by AP).

In another recent work [9], Graf investigated the va-lidity of AP as originally state by Archimedes (287-212b.C.) in his On Floating Bodies, Book I [10]. For abody less dense than the liquid, treated in Propositions4 to 6, Graf noted that the term ‘buoyant force’ does

not appear, but in Proposition 6, namely [10]

If a solid lighter than a fluid be forcibly im-mersed in it, the solid will be driven up-wards by a force equal to the difference be-tween its weight and the weight of the fluiddisplaced,

an upward force is mentioned which, in modern words,would be the apparent weight. He also points out thatthe proofs by Archimedes were only for static cases inwhich there is fluid under the body, so our ‘bottom case’was overlooked. It is only for a body denser than the liq-uid, treated in Proposition 7, that Archimedes speaksof a body in contact to the bottom, but he again men-tions only the apparent weight. Graf then devises amethod to measure the apparent weight directly witha waterproof scale, establishing that this weight is thesame whether or not there is liquid under the block.For a block less dense than the liquid, however, he ar-gues that “the apparent weight would be negative andprovision would have to be made to keep the object incontact to the scale.”

By suspecting that the above conclusion by Grafcould be wrong, we have developed a simple experimentto test it. Since the experiment by Jones and Gordonrequires the complement by Ray and Johnson [5], whichdemands a vacuum pump (not available in most schoolsand undergraduate labs), here in this work we presenta simpler, cheaper experimental setup for checking thedirection of the buoyant force acting upon a block ofwood immersed in water and touching the bottom ofan aquarium.

2. Basic theory for the ‘bottom case’

Let us start by defining the buoyant force as the netforce that a fluid exerts on the part of the surface ofa body that is effectively in contact to the fluid.3 For asolid body with a volume V fully submerged in a liquidwith a density ρ, as we are interested here, AP predictsan upward force with a magnitude

Fb = ρ V g , (1)

i.e., the weight of the liquid displaced by the body, gbeing the local acceleration of gravity [1, 2, 11]. Notethat, according to AP, the BF should not vary withdepth.

For a rectangular block resting on the bottom of acontainer filled with a liquid, however, given that theblock is fully submerged and assuming that no liquidseeps under the block, the horizontal forces cancel outand the net force exerted by the liquid is expected topoint downward, as indicated in Fig. 1. In Ref. [13],

3This differs from the definition adopted by Mungan in a recent work [12]. There, in order to keep the direction of the BF upwardin all static cases, he proposes that the BF could be defined as the negative of the weight of the fluid displaced by the body. For a bodyfully surrounded by a liquid, both definitions are equivalent.

A downward buoyant force experiment 2309-3

the first author predicts a magnitude p top A for thatforce, p top being the pressure at the level of the top ofthe block, whose area is A. From Stevinus law [1], thisyields

Fb = − [ p0 + ρ g (H − h) ]A k , (2)

where k is the unit vector pointing upwards, p0 is theatmospheric pressure, H is the height of the liquid col-umn above the bottom of the block, and h is the heightof the block, as indicated in Fig. 1. Since the liquidpressure in any point of the bottom is pb = p0 + ρ gH,one can write the force in Eq. (2) as Ref. [13]

Fb = − (pb A− ρ V g) k , (3)

where V is the volume of the block. In our experiment,a technique similar to that introduced in Ref. [8] is em-ployed in order to reduce the liquid seepage under theblock, as indicated in Fig. 2. While no liquid seeps inthe rectangular hole, the net hydrostatic force exertedon the block will be4

Fb = − (ptop A− p0 Aair − pb Aseep) k , (4)

where Aair (< A) is the area of the portion of the basethat is in contact to air and Aseep (≪ A) is the area ofthe bottom of the block in contact to liquid under theblock.5 If no liquid seeps under the block, then Eq. (4)yields a downward force given by

⌋

Fb = ptop A− p0 Aair = p0 A+ ρ g (H − h)A− p0 Aair = ρ g H A− ρ g V + p0 (A−Aair) . (5)

More generally, if only a fraction 0 < f < 1 of the region between the rectangles indicated in Fig. 2 (whose area isA−Aair) is in contact to the liquid, then the net force exerted by the fluids on the block is6

Fb = − [ptop A− p0 Aair − pb f (A−Aair)] k

= − [ρ g (H − h)A+ p0 A− p0 Aair − f (p0 + ρ g H) · (A−Aair)] k

= − [(1− f) ρ g HA+ (1− f) p0 (A−Aair) + f ρ g HAair − ρ g V ] k .

(6)

⌈

From this result, one easily deduces that a downwardBF arises whenever

H > Hmin =V − (1− f) (A−Aair) p0/(ρg)

A− f (A−Aair), (7)

where Hmin is the depth for which the force in Eq. (6)is null. Let us check the limits f → 0 and f →1. For f → 0, Fb is that given in Eq. (5), henceHmin = h − (1−Aair/A) p0/(ρ g) . Since Aair < A,this height is of course less than h, which means that,being the block fully submerged, the force points down-ward. For f → 1, on the other hand, Fb reduces toρ g (Aair H − V ) and then

Hmin =V

Aair=

A

Aairh . (8)

In this case, Hmin > h and depths H ≤ Hmin mustbe avoided in seeking for a downward BF.

Aair

A

Figure 2 - Our technique for reducing the contact between thebase of our block and the liquid under the block. In a rubberpad, we cut a rectangular hollow with an area Aair somewhatsmaller than the area A of the base of the block, indicated bythe dashed rectangle.

3. Our experiment

We have developed a simple, low cost experiment to in-vestigate the net force exerted by water on a block ofwood immersed in it and resting on the bottom of an

4Since p0 ≈ 105 N/m2, this downward force can be strong enough to make it difficult to detach the block from the bottom. Thismakes it dangerous, e.g., to step on the drain of a deep tank or swimming pool when it is open for emptying.

5Note that Aseep = A only at the end of each run, when the liquid wets the base of the block and moves it upwards.6At this point, we are assuming the liquid seeps under the block but does not touch the part of its bottom which is just above the

hollow.

2309-4 Lima et al.

aquarium. Our experiment allows for checking whetherthat force is directed upward, as predicted by AP, ordownward, as predicted in Ref. [13].

Our block is a rectangular one made of wood,with a mass of 229.6 grams and dimensions14.0 cm × 7.0 cm × 3.3 cm (hence a density of0.7 grams/cm3). It was put on a rubber pad with arectangular hollow whose length and width are 0.5 cmnarrower than the corresponding quantities in the baseof the block (namely, 13.5 cm × 6.5 cm), as illustratedin Fig. 2. The pad has a thickness of 5 mm. Thistechnique is similar to that introduced by Bierman andKincanon in Ref. [8]. Though water slowly seeps underthe block due to small roughnesses of both the woodand rubber surfaces, the part of the base of the blockjust above the hole, whose area is Aair = 87.75 cm2

(thus smaller than A = 98 cm2), has initially no wa-ter touching the block (only air). As a consequence,the hydrostatic pressure of water acts only on the smallarea A − Aair = 10.25 cm2, which is not sufficient forpushing the block up. We have used a plastic bottleconnected to a hose to slowly fill the glass box with tapwater (density ρ = 1.00 g/cm3), as shown in Fig. 3.In order to avoid the block disturbance and reduce thewater seepage, we slightly pressed down the top of theblock with one of our fingers, but only until the water-line to attain the depth H = Hmin ≈ 4 cm, accordingto Eq. (8). For H > Hmin the water itself presses theblock down and we could safely remove the finger. In allruns, the block remained at the bottom for more thana minute, until water to fulfill the hollow under theblock, after which it floated. This crucial experimentfor AP shows the presence of a ‘buoyant’ force pressingthe block down during all time prior to its detachmentfrom the bottom of the aquarium.

On presenting this counterintuitive experiment toour first-year students, we have noted that most of themsuspected of some ‘trick’, i.e. that we could be usingsome other source of downward force. This revealedtheir trust in AP, but we ruled out these suspects bypositioning the glass box in a manner to leave the partbelow the block out of the table, as shown in Fig. 3,which allowed them to see the water seeping under theblock. In fact, some students asked us to examine theblock of wood, seeking for a magnet or some other trick.They also suspected that the block could be ‘heavier’than water, which was ruled out at the end of the exper-iment, when the hollow was finally occupied by waterand the block was pushed upwards until floating.

4. Conclusions

We have drawn the readers attention to the fact thatthe upward force predicted by AP is not always equalto the hydrostatic force exerted by a liquid on a bodyimmersed in it, as observed in our experiment with ablock of wood in contact to the bottom of an aquarium,

in which the force Fb stated in Eq. (2) points down-ward for all H > Hmin, contrarily to AP. Our result forthe BF agrees to Bierman and Kincanon experiment, inwhich the force exerted by the liquid in a block denserthan it is shown to point downward [8]. It would not besurprising that Archimedes, one of the greatest geniusesof the ancient world, had enunciated his original propo-sitions with remarkable precision and insight, howeverthere are some instances which he did not realize. Oneof these instances is shown here to be an exception toAP. We suggest the inclusion of this experiment as aninteresting complement to the apparent weight experi-ment routinely reproduced by students in introductoryphysics labs.

Figure 3 - One of us (GTB) using a pet bottle and a hose to(slowly) fulfill the glass box with water. The hose enters the boxfrom a distance of the block, in view to avoiding any disturbancein the block equilibrium and reducing the liquid seepage. Note,at the left, the vertical rule used for measuring H.

References

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[4] G.E. Jones and W.P. Gordon, Phys. Teach. 17, 59(1979).

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A downward buoyant force experiment 2309-5

[6] B.M. Valiyov and V.D. Yegorenkov, Phys. Educ. 35,284 (2000).

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spring2006/mungan.html.

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